Around a conjecture of P. Hall

In the paper, we discuss perspectives of future investigations of the Hall $\pi$-properties $E_\pi$, $C_\pi$ and $D_\pi$ in finite groups. A series of open problems is stated, both comparatirely new and well-known ones. It is proven that there are infinitely many infinite sets $\pi$ of primes with $E_\pi\Rightarrow D_\pi$. Precisely if $\pi$ consists of the primes $p>x$, for every real $x\ge7$ then $E_\pi\Rightarrow D_\pi$. This result continues the investigations initiated by well-known Hall's conjecture of 1956 that $E_\pi\Rightarrow D_\pi$ for every set $\pi$ of odd primes. This conjecture was disproved by F. Gross, who showed in 1984 that, for every finite set $\pi$ of odd primes with $|\pi|\ge2$, there exists a finite group $G$ such that $G\in E_\pi$ and $G\notin D_\pi$.